Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $\kappa$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $\kappa \ge 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green’s functions along certain paths.